Vectors in Two or Three Dimensions

Ann Hirst

• Publisher: Butterworth-Heinemann
• Year: 1995
• Print ISBN: 9780340614693
• eBook ISBN: 9780080572017
• Topic: Mathematics
• Subtopic: Linear Algebra
• Index: Mathematics

Vectors in 2 or 3 Dimensions provides an introduction to vectors from their very basics. The author has approached the subject from a geometrical standpoint and although applications to mechanics will be pointed out and techniques from linear algebra employed, it is the geometric view which is emphasised throughout.Properties of vectors are initially introduced before moving on to vector algebra and transformation geometry. Vector calculus as a means of studying curves and surfaces in 3 dimensions and the concept of isometry are introduced later, providing a stepping stone to more advanced theories.* Adopts a geometric approach* Develops gradually, building from basics to the concept of isometry and vector calculus* Assumes virtually no prior knowledge* Numerous worked examples, exercises and challenge questions

1

Introduction to Vectors

1.1 Vectors and scalars

When people ask‘What is a vector?’ it is as difficult to answer as‘What is a number?’ Both vectors and numbers are abstract ideas which represent more concrete quantities. We start by learning that two apples added to two apples gives us four apples, two pencils added to two pencils gives us four pencils, and so on using physical objects, and it is some time before we link this with the more abstract concept 2 + 2 = 4. With a vector there are two quantities involved in the representation, and we generally think of these as magnitude and direction, and we often use the term length as an alternative for magnitude. So a vector is defined as something having both magnitude and direction, and anything which has just a magnitude attached to it is called a scalar. In this book all our scalars will be real numbers, but readers should be aware that there are vector spaces for which the scalars are complex numbers or even more exotic beings.

One way of differentiating between vectors and scalars is by considering the difference between the distance between two points, which is a scalar, and the displacement of one point from another, which is a vector, and which we can regard as what we have to do to get from one point to another. In this case we need to know not only how far we have to go, but also in which direction. Buckingham Palace is 1.25 km from Trafalgar Square, but if someone is starting from Trafalgar Square and wishes to get to Buckingham Palace, it is no good walking 1.25 km to the east!

Examples of scalar quantities are distances, speeds and masses, and examples of vector quantities are displacements, velocities, weights.

Notation

We shall represent vectors in bold type and scalars will be written in italics, so v represents a vector, but s represents a scalar. The vector which represents displacement from a point A to a point B will be written as AB, and the vector from B to A as BA. This indicates how vital it is to make the direction clear on a, diagram, and we shall use arrows to indicate direction, as in Fig 1.1.

Fig 1.1

1.2 Basic definitions and notation

This book is concerned mainly with vectors in two or three dimensions. From a fairly early stage we are used to dealing in two dimensions by choosing two axes of coordi...

Table of contents

• Cover image
• Title page
• Table of Contents
• Inside Front Cover
• Copyright
• Series Preface
• Preface
• Chapter 1: Introduction to Vectors
• Chapter 2: Vector Equation of a Straight Line
• Chapter 3: Scalar Products and Equations of Planes
• Chapter 4: Vector Products
• Chapter 5: The Vector Spaces ℝ2 and ℝ3, Linear Combinations and Bases
• Chapter 6: Linear Transformations
• Chapter 7: General Reflections, Rotations and Translations in ℝ3
• Chapter 8: Vector-valued Functions of a Single Variable
• Chapter 9: Non-rectangular Coordinate Systems and Surfaces
• Answers to Exercises
• Index